Moderate -0.3 This is a straightforward binomial expansion question requiring students to identify which term has x^0 (the constant term) and calculate its coefficient. While it involves a two-step process (finding the correct term index, then computing the coefficient), it's a standard textbook exercise with no conceptual difficulty beyond basic binomial theorem application.
Identifying term as \(20(2x)^3\left(\frac{5}{x}\right)^3\) oe
M3
Condone lack of brackets. M1 for \([k](2x)^3\left(\frac{5}{x}\right)^3\) soi (e.g. in list or table), condoning lack of brackets. And M1 for \(k = 20\) or e.g. \(\frac{6\times5\times4}{3\times2\times1}\) or for \(1\ 6\ 15\ 20\ 15\ 6\ 1\) seen (Pascal's triangle, even if no attempt at expansion). And M1 for selecting the appropriate term. \(x\)s may be omitted; e.g. M3 for \(20\times8\times125\). M0 for binomial coefficient if it still has factorial notation
\(20000\)
A1
Or B4 for 20000 obtained from multiplying out \(\left(2x+\frac{5}{x}\right)^6\). Allow SC3 for 20000 as part of an expansion
[4]
## Question 6:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Identifying term as $20(2x)^3\left(\frac{5}{x}\right)^3$ oe | M3 | Condone lack of brackets. M1 for $[k](2x)^3\left(\frac{5}{x}\right)^3$ soi (e.g. in list or table), condoning lack of brackets. And M1 for $k = 20$ or e.g. $\frac{6\times5\times4}{3\times2\times1}$ or for $1\ 6\ 15\ 20\ 15\ 6\ 1$ seen (Pascal's triangle, even if no attempt at expansion). And M1 for selecting the appropriate term. $x$s may be omitted; e.g. M3 for $20\times8\times125$. M0 for binomial coefficient if it still has factorial notation |
| $20000$ | A1 | Or B4 for 20000 obtained from multiplying out $\left(2x+\frac{5}{x}\right)^6$. Allow SC3 for 20000 as part of an expansion |
| **[4]** | | |
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6 The binomial expansion of $\left( 2 x + \frac { 5 } { x } \right) ^ { 6 }$ has a term which is a constant. Find this term.
\hfill \mbox{\textit{OCR MEI C1 2013 Q6 [4]}}